Q13/038) Evaluate the following limit without using calculus:

(2−√(2+x))/((2^(1/3)−(4−x)^(1/3))

Solution: 

I would rationalize the denominator to get

(2−√(2+x))(2^(2/3) + 2^(1/3)  (4-x)^(1/3) + (4-x)^(1/3)) /( 2 – (4-x))

Now 2- (4-x) = - ( 4- (2+x) = - (2−√(2+x))( (2+√(2+x))

So we get - (2^(2/3) + 2^(1/3)  (4-x)^(1/3) + (4-x)^(1/3))/ ((2+√(2+x))

Putting x =2 we get - (2^(2/3) + 2^(2/3) + 2^(2/3))/( 2 + 2) = - 3 * 2^(2/3) / 4